Nonrelativistic Calculation

Relativistic quantum mechanics yields the same type of expressions for the isomer shift as the classical approach described earlier. Relativistic effects have to be considered for the calculation of the electron density. The corresponding contributions to i/ (0)p may amount to about 30% for iron, but much more for heavier atoms. In Appendix D, a few examples of correction factors for nonrelativistically calculated charge densities are collected. Even the nonrelativistically calculated p(0) values accurately follow the chemical variations and provide a reliable tool for the prediction of Mossbauer properties [16]. 

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. 

An even simpler approach to relativity, for heavy elements, is to use effective core potentials (ECPs) to represent the core electrons, taking the potentials from various compilations in the literature that explicitly include relativistic effects in the generation of the ECPs. References to such ECPs are given by Dyall et al. [103]. These relativistic ECPs (RECPs) allow the inclusion of some relativistic effects into a nonrelativistic calculation. Since ECPs will be treated in detail elsewhere, we will not pursue this approach further here. We may note, however, that recent comparisons with Dirac-Fock calculations suggest that the main weakness in the RECPs is not the treatment of relativity but the quality of the ECPs themselves [103]. Different RECPs gave spectroscopic constants with a noticeable scatter, compared to Dirac-Fock, but the relativistic corrections (difference between an RECP and the corresponding ECP value) were fairly consistent with one another. 

The lesson we learn from these considerations is the need to understand that elaborate nonrelativistic calculations of dipole-supported resonances lying within AE of the corresponding nonrelativistic threshold Eth are mere model calculations that result in what are quite far from the actual physical system. 

All-electron nonrelativistic calculation, at CISD/TZP(f,d) from Reference 142. 

The Sn 5 s and 5p radial functions, from a nonrelativistic calculation for the free 5sz5pz atom, are plotted in Fig. 7. Roughly 8% of the 5s charge extends outside the Wigner-Seitz radius, rws, for / —Sn the 5s orbital, with much of its density in a region in which Zen is about equal to the valence, is actually somewhat in the interior of the atom. It is not unlike the d orbitals of transition metals, which, as earlier noted, maintain much of their atomic quality in a metal. Thus it is quite plausible that the valence s character in Sn is much like the free atom 5 s, except for a renormalization within the Wigner-Seitz cell. The much more extended 5p component, on the other hand, is not subject to simple renormalization the p character near the bottom of the band takes on a form more like the dot-dash curve of Fig. 7. It nevertheless appears useful to account for charge terms of a pseudo P component and a renormalized s. 

Table 2 shows the relativistic and nonrelativistic values of Pb for the diatomic and hexafluoride molecules. For the diatomic molecules, it was found that the absolute values of APp (the Pb between the relativistic and nonrelativistic calculations) increases with atomic-number Z. 

Using pseudopotentials has several major beneficial consequences (i) Only the valence electrons must be treated explicitly, thus the number of equations to be solved [Eqs. (13)] can be reduced drastically (ii) the pseudoorbitals are very smooth near the atomic core, and thus Tout can be reduced drastically and (iii) important relativistic effects of the core electrons of heavy elements such as the 5d elements can be included in nonrelativistic calculations. The major downsides are that the potential v(r) in Eq. (3) must be replaced with a more complicated and computationally expensive nonlocal pseudopotential and, more importantly, that the transferability of the pseudopotential, i.e., its accuracy in different bonding environments, may not be perfect. Developing highly transferable pseudopotentials that can be used at as low an cut as possible is a major current topic of research. 

The appropriately chosen electron-nucleus potential is denoted as Vnuc(0- Usually, we shift the energy scale with /T = /3 — 1 to yield energy expectation values which are directly comparable with those obtained from nonrelativistic calculations. 

Within a nonrelativistic calculation of the hyperfine fields in cubic solids, one gets only contributions from s electrons via the Fermi contact interaction. Accounting for the spin-orbit coupling, however, leads to contributions from non-s elections as well. On the basis of the results for the orbital magnetic moments we may expect that these are primarily due to the orbital hyperfine interaction. Nevertheless, there might be a contribution via the spin-dipolar interaction as well. A most detailed investigation of this issue is achieved by using the proper relativistic expressions for the Fermi-contact (F), spin-dipolar (dip) and orbital (oib) hyperfine interaction operators (Battocletti... 

The errors (line (d)) due to the use of the retarded non relativistic formulas, with respect to the retarded relativistic ones, are not very large for E — me2 < 1,300Kev (Z = 1), but then they increase fastly, and become considerable for E very large. That shows the profound difference between the relativistic and the nonrelativistic calculations. One can notice on Table 12.1 incidence of the value of Z on these errors. 

We recall that the nonrelativistic calculation cannot be applied in the cases of degeneracies (for reasons analog to the differences between the normal and anomal Zeeman effect), as it is schown by the formulas (9.20)-(9.29). 

At least the study of the term Wo will allow us to show the difference of the values of this term between those obtained by the relativistic and nonrelativistic calculations. This difference is weak for the contribution of the discrete spectrum and the low levels of the continuum but becomes considerable (see Note below) for the ones of the continuum of high levels and explains the necessity of the mass renormalization represented by the term Wm-... 

In the nonrelativistic calculation the Dirac equation is replaced by the Schrodinger one. The formula that is obtained (see [2]), which is convergent, is, if the dipole approximation is applied (i.e. Tj-(k) are replaced by Tj-(O)), the formula used in [4] for the first calculation proposed for the explanation of the Lamb shift. But this last formula is divergent and its use implies that the integration upon k is cut off for a k = kmax. In [23] the value of kmax = ante2 has been proposed and was used in the following calculations of the Lamb shift. 

These values are not very different from the ones (see below) obtained in the relativistic calculation. But when the contribution of the continuum is taken into account especially for the levels of energies greater than amc2 the divergence is such that all comparison between the relativistic and the nonrelativistic calculation is to be abandoned (see Note below). 

G-spinors satisfy the analytic boundary conditions (137) for jc < 0 and (138) for tc > 0. A G-spinor basis set consists of functions of the form of (147-149) with suitably chosen exponents Xm, m = 1,2,..., d - The choice of sequences Xfn which ensure linear independence of the G-spinors and a form of completeness is discussed in [86]. It is often sufficient to use the GTO exponents from nonrelativistic calculations, of which there are many compilations in the literature perhaps augmented with one or two functions with a larger value of A to improve the fit around the nucleus. 

Basis sets taken from nonrelativistic calculations are a good first choice for most problems since they provide a good representation of the valence electron distribution. However, we know that the shape of the wavefunction is sensitive to the nuclear charge close to the nucleus, and it is therefore important to choose basis functions for the most penetrating s and p orbitals to improve the fit at short range. The simplest way to do this is to add some larger exponents. 

D. Andrae, J. Hinze, Numerical Electronic Structure Calculations for Atoms. I. Generalized Variable Transformation and Nonrelativistic Calculations, Int. J. Quantum Chem. 63 (1997) 65-91. 

It is much more straightforward to interpret relativistic effects in terms of PT rather than from a comparison of relativistic and nonrelativistic calculations. 

Relativistic effects in a certain property may be defined as the difference between the results obtained from a relativistic and a nonrelativistic calculation. Clearly, using this definition the quantitative study of relativistic effects will depend on the choice of the (relativistic) Hamiltonian as well as on the qual-... 

We will deal only with computational procedures that are normal for molecnles encoimtered in organic photochemistry. These methods depend heavily on the assumption that the spin-orbit conpling term is only a minor perturbation. In snch an instance, it is common to not include small relativistic terms such as spin-orbit coupling in the Hamiltonian from the beginning bnt to include them as an afterthought after an ordinary nonrelativistic calculation. This is usually done nsing pertnrbation theory or response theory. 

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